Are you talking about these cases?
Pruneyard Shopping Center v. Robins, 447 U.S. 74 (1980)
Lloyd Corp. v. Tanner, 407 U.S. 551 (1972)
Pruneyard, the SCOTUS discussed California's affirmative right to Free Speech, a right which is different to the US Constitution's prohibition against restricting Speech.
In American constitutional law, this case established two important rules:
under the California Constitution, individuals may peacefully exercise their right to free speech in parts of private shopping centers regularly held open to the public, subject to reasonable regulations adopted by the shopping centers
under the U.S. Constitution, states can provide their citizens with broader rights in their constitutions than under the federal Constitution, so long as those rights do not infringe on any federal constitutional rights
This holding was possible because California's constitution contains an affirmative right of free speech which has been liberally construed by the Supreme Court of California, while the federal constitution's First Amendment contains only a negative command to Congress to not abridge the freedom of speech.
By contrast, In the Tanner case, the SCOTUS says that there is no right to free speech on private property
Justice Lewis F. Powell concluded that the respondents could have distributed their handbills on "any public street, on any public sidewalk, in any public park, or in any public building." Therefore, respondents were not entitled to exercise their free-speech rights on the privately owned shopping-center property.
It should be added that even in the California Constitution case in Pruneyard, there were quite a few permissible limitations on Speech, including disclaimers, warnings against the content of the views being expressed and confining the speech to certain areas of the mall. The California law itself continued to be narrowed and many exceptions were added that basically relegated the right to free speech in California malls to the food courts.
For example, this "Please do not contribute to solicitors" sign:
Not exactly how Pruneyard's view of Free Speech would be applied to Twitter because although the States can have broader free speech rights, Twitter's influence isnt confined to one State and therefore a California citizen's speech would interfere with a NYers right not to be subject to it.
Congress almost certainly cant do anything about this either. Frankly, if Twitter wanted to come out and say, we dont like right wing speech and it's off, I dont think Congress could do anything about it.
However, this recent concept that anything but unfettered, extremist speech which can include unfounded conspiracy theories and outright lies is an exercise in bias or persectuion is truly disturbing. It is also an admission by extremist groups that they are incapable of establishing their own channels of communication and thus must hijack, existing, popular platforms in order to disseminate their views. In addition, one could make the argument that when Twitter allows these views to be aired, these extremist groups profit from Twitter's moderate and commercial appeal to make it seem like extremist views are much truer than they actually are. In other words, crackpots are counting on the public's response being "If it's on a trustworthy site like Twitter, it must be true.
This is why private property is rarely subject to any obligation to promote free speech or stopped from editing it.
This second part about bias leading to the erosion of libel protection for the Press is a complicated one. But the concept of bias seems to carry its own bias. In any case, this new conservative mantra of bias is flawed for several reasons:
1. It's only a problem because social conservatives feel under siege. Bias wasnt a problem when they felt in control; rather, it was considered being right-thinking.
1A. I dont see groups yelling about Bias when they think the bias is in their camp or expounds views they tend to believe or want to believe. This seems especially true with conservative media which cut and pastes or tinkers with reality to try to fit an uninterruptible fantasy narrative. Even though he also acknowledged the claims were false, Rush Limbaugh recently supported a certain Orangey guy's false claims about a long debunked murder as being playful and trying to start fires. Is this a case of bias? Should he be sued for libel? I didn't hear conservative voices getting outraged over that. Be careful that a claim about bias isnt itself steeped in bias.
2. Bias isnt necessarily a problem. Corruption might be but not bias. If we hire people to investigate and arrest some universally detested group, we usually dont complain if the investigators are biased. We would draw the line at the fabrication of evidence but not, for instance, if the prosecutors hated organized crime; we might even consider that a positive attribute.
3. "Bias" is the new secret weapon for extremists who are restrained from making sweeping and, presumably, influential claims with no support or to condemn a source if they believe it to be detrimental to their viewpoint.
In any case, Bias as a form of unfair limitation can only exist if the original claim being edited is in fact true and many of those original claims carry biases of their own, especially when they come from a dark place or are unsupported by facts/evidence.
Someone publishing a story such as children steal mail in election ballots as if it is a common and long standing problem when in fact it is supported by no evidence and may itself emanate from the original claimants desire to influence an election and which is then debunked or deleted by social media cannot go on to legitimately claim "bias" solely because they arent free to construct any sort of reality they desire.
The Fox News Seth Rich case is one of the few recent cases where a Media company has gone too far with promoting a conspiracy it knew to be false in order to help a political party. This case has yet to be decided but if FOX should lose this case, it might actually mandate that the Press and social media be a lot more careful with what sources and views they allow which ironically for a lot of extremist groups will seem like even more "bias" because all sorts of stories will have to either be verifiable or get deleted.
However, i dont think that bias is a reason being considered as a reason to open up libel liability for the Press or social media.